Problems with Fermions

One hundred years ago, Enrico Fermi showed how the Pauli exclusion principle fundamentally changes the statistical behavior of a quantum gas. Since then, fermions have become central to our description of matter.

Yet systems composed of many interacting fermions remain among the most difficult quantum systems to describe theoretically.

Part of this difficulty is rooted in fermionic statistics itself. Antisymmetry gives rise to wave functions with complicated nodal structure and, in many numerical approaches, to the notorious fermion sign problem. At the same time, the microscopic interaction may itself be highly nontrivial. In nuclear physics, chiral effective field theory provides a systematic low-energy framework for constructing nuclear forces consistent with the low-energy symmetries of QCD, yielding Hamiltonians that can then be tackled with many-body methods.

There are also remarkable ways in which the complexity of interacting fermions can simplify. Landau's Fermi-liquid theory showed that, at sufficiently low energies, a broad class of interacting Fermi systems can be described in terms of quasiparticles and a small set of effective parameters.

I will discuss how these different layers of difficulty shape modern nonperturbative approaches to fermionic systems, focusing on quantum Monte Carlo methods and density functional theory. Examples will be drawn from atomic nuclei, neutron matter and neutron-star matter, and ultracold Fermi gases. Despite their vastly different scales, these systems share closely related many-body physics and provide complementary laboratories for understanding strongly interacting fermionic matter.